V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 1

A dynamic route choice model

for public transport networks with boarding queues

Valentina. Trozzi1, Ioannis Kaparias

2, Michael Bell

3 and Guido Gentile

4

1 Department of Civil and Environmental Engineering, Imperial College London,

v.trozzi09@imperial.ac.uk

3 School of Engineering and Mathematical Sciences, City University London,

kaparias@city.ac.uk

3 Institute of Transport and Logistics Studies, University of Sydney,

michael.bell@sydney.edu.au

4 Dipartimento di Ingegneria Civile Edile e Ambientale, Università di Roma “La

Sapienza”,

guido.gentile@uniroma1.it

Abstract

The concepts of optimal strategy and hyperpath were born within the framework of

static frequency-based public transport assignment, where it is assumed that travel times

and frequencies do not change over time and no overcrowding occurs. However, the

formation of queues at public transport stops can prevent passengers from boarding the

first vehicle approaching and can thus lead to additional delays in their trip. Assuming

that passengers know from previous experience that for certain stops/lines they will

have to wait for the arrival of the 2nd, 3rd, ..., k-th vehicle, they may alter their route

choices, thus resulting in a different assignment of flows across the network. The aim of

this paper is to investigate route choice behaviour changes as a result of the formation

and dispersion of queues at stops within the framework of optimal travel strategies. A

new model is developed, based on modifications of existing algorithms.

Keywords: transit networks, dynamic hyperpaths, passenger congestion, FIFO queues,

Erlang distribution

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 2

1. Introduction

Investments in new public transport infrastructure and services may be

financially constrained and, thus, sound transit assignment models are needed for

describing and predicting passengers’ flows and justifying resource allocation in

transport planning. An open question in this research realm is how to deal with recurrent

overcrowding that may lead to passengers changing their route and mode choice, time

of departure and sometimes even their final destination. Consequently, a step forward in

the public transport modelling may be achieved by developing a new route choice

model, as presented in this work, which is capable of considering: travel time

variability, formation and dispersion of passenger queues at stops and the resulting

increase in waiting times, in densely connected transit networks. On the other hand,

although the effects on departure time choice and mode choice are crucial for the

development of a multi-modal and dynamic assignment procedure, they are out of the

scope of the present study.

The application of this route choice model is twofold. On one hand it can be

embedded in a dynamic assignment model to capture the formation and dispersion of

passenger queues at transit stops and evaluate the resulting increase in waiting times.

On the other hand, when queues length is estimated by means of transit assignment

models, it can benefit dynamic journey planners, enabling them consider congestion

patterns on the transit network.

The rest of the paper is organised as follows. In next section, the background

research is presented, while the methodology is explained in Section 3. In Section 4,

numerical examples will be presented and conclusions will be drawn in Section 5.

Details about the solution algorithm are provided in the Appendix 1.

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 3

2. Background research

Much of previous research on route choice in public transport networks has focussed on

static conditions, where it is assumed the relevant model variables, such as travel times

and line frequencies, are fixed, and passengers can always board the first approaching

carrier. In this context, if no service schedule is published or services are highly

frequent and/or unreliable, users have no explicit knowledge about carriers’ arrival time

at transit stops. So, if there are two or more competing lines, so-called common lines

(Chriqui and Robillard 1975), they may face the question if it is more convenient to

board the first approaching line or wait for another, they consider more convenient.

Some authors (Spiess 1983, Spiess 1984, Nguyen and Pallottino 1988, Spiess

and Florian 1989) prove that, when there is this uncertainty, rather than the single

shortest itinerary between origin and destination, route choice can be modelled as the

selection of the optimal strategy (Spiess and Florian 1989), and graphically represented

as the shortest hyperpath on an oriented hypergraph1 modelling the transit network

(Nguyen and Pallottino 1988, Wu et al. 1994, Nguyen et al. 1998).

The optimal strategy is selected pre-trip and, starting from the origin, involves

the iterative sequence of: walking to a public transport stop or to the destination,

selecting the attractive (Nguyen and Pallottino 1988) lines to board and, for each of

them, the stop where to alight. Once travelling towards the destination, if two or more

attractive lines are available at a stop, the best option is to board the first approaching

(Spiess 1983, 1984).

It is well known (Billi et al. 2004, Nökel and Wekeck 2007) that this definition

of optimal travel strategy only applies when:

1 For a detailed review of directed hypergraphs and their applications, we refer the reader to

Gallo et al. (1993).

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 4

passengers have no explicit knowledge about the carrier’s arrival times at transit

stops and the available capacities of arriving carriers;

the vehicle arrivals of different lines at the stop are not synchronized, and, for

each line, follow a Poisson distribution;

passenger arrivals at stops are not timed to coincide with vehicle arrivals; and

passengers try to minimize their total expected travel time to their destinations.

As such, formation and dispersion of passenger queues are not captured, and the

dependency of waiting time and passengers’ distribution among the different attractive

lines from overcrowding is not considered. Recurrent passenger congestion is one of the

major problems faced by large city public transport networks and the distortions

brought about by neglecting this phenomenon can be significant, as explained by the

following example.

Consider a small network with two nodes, origin and destination, and two lines

that have the same travel time to destination upon boarding (10 minutes). As detailed in

Table I, Line 1 is more frequent but is also congested and passengers cannot board the

first vehicle approaching.

Table I: Line 1 and Line 2 congestion levels, average headways and travel time upon

boarding. Line 1 is congested and passengers are not able to board the first vehicle.

Average

frequency [min-1

]

Travel time to destination

upon boarding [min]

Congestion level

Line 1 1/3 10 Congested

Line 2 1/6 10 Not congested

If congestion is disregarded, passengers’ distribution (i.e. the probability to

board Line 1 or Line 2) and total expected waiting time solely depend on the average

frequencies of the services, as in (Nguyen Pallottino 1988, Spiess and Florian 1989),

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 5

and their values are shown in Table II. The results are clearly distorted because,

although congested, Line 1 attracts the major percentage of passengers and its

congestion would become even more severe. Thus, if embedded in an assignment

procedure, this route choice would not lead to equilibrium conditions.

Table II: Line 1 and Line 2 boarding probability and total expected waiting time at the

considered stop.

Line 1 Line 2 Origin Stop

Boarding probability 0.67 0.33 -

Total Expected waiting time

[min] - -

2

Most of the research carried out to overcome this shortcoming in the context of

dense public transport networks, where common lines may be available from the same

stop, has dealt with overcrowding in case of passengers mingling at the stop (De Cea

and Fernandez 1993, Marcotte and Nguyen 1998, Cominetti and Correa 2001, Kurauchi

et al. 2003, Cepeda et al. 2006, Schmöcker et al. 2008, Leurent et al. 2011, Leurent and

Benezech 2011). This behavioural assumption implies that no waiting priority is

respected and, in case of oversaturation, all passengers waiting at a stop have the same

probability to board the next carrier to approach (provided the carrier is attractive). Thus

a possible simple solution (De Cea and Fernandez 1993) would consider the effective

frequency, namely the line frequency perceived by waiting passengers that decreases as

the probability of not boarding its first arriving carrier increases. If the stop in the

previous example is considered and we assume the ‘fail-to-board’ (Schmöcker et al.

2008) probability for Line 1 is 0.5, while passengers are always able to board a carrier

of Line 2, then effective frequencies, passengers’ distribution and total expected waiting

time are those displayed in Table III. The share of Line 1 decreases of about 30% in

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 6

favour of Line 2 and, as expected, the consideration of overcrowding leads to an

increase of the total expected waiting time.

Table III: Effective frequency and boarding probability of Line 1 and Line 2; total

expected waiting time at the considered stop.

Line 1 Line 2 Origin Stop

Effective frequency [min-1

] 1/6 1/6 -

Boarding probability 0.5 0.5 -

Total Expected waiting time

[min]

- - 3

For stations and stops with large platforms, it is acceptable to assume passengers

mingle and apply results from the afore-mentioned studies. However, in urban bus

networks the stop layout is usually such that, when passenger congestion occurs, users

arriving at the stop would join a FIFO (first-in first-out) queue and board the first line of

their attractive set that becomes available. Models based on the mingling queuing

protocol are clearly not applicable to the latter scenario. Early attempts to overcome this

shortcoming are made by Gendreau (1984) and Bouzaïene-Ayari (1988) by using a bulk

queue model, but its complexity prevents practical applications of the models. In a latter

study, Leurent and Benezech (2011) developed a model based on the ‘attractivity

threshold’, but in this case rather than assuming the route is chosen pre-trip, authors

consider a completely adaptive behaviour at transit stops.

On the other hand, we prove that in the context of commuting passengers, who

know by previous experience the number of carrier passages they must let go before

being able to board, expected waiting times and passenger distribution among attractive

lines may be calculated by extending the formulas presented by Nguyen and Pallottino

(1988) and Spiess and Florian (1989). Results obtained with our model will be

compared with those detailed in Table II (uncongested model) and Table III (effective

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 7

frequency model), and it will be shown that the FIFO queue model penalises the

congested line, in terms of passengers’ distribution, more than the uncongested and the

effective frequency model, while the total expected waiting time increases.

Prior to that, the proposed route choice model will be introduced in the next

section, together with the mathematical framework and the graph representing the

transit network.

3. Methodology

3.1 Network model: notation and definitions

The transit network comprises a set of lines ℑ ⊆ ℵ (ℵ indicates the set of natural 140

numbers) and, together with the pedestrian network, it is represented by a directed

hypergraph (Gallo et al. 1993) HG = {N, E}, where N = {i | i = 1, 2, …, n} is the node

set and E = {e | e = 1, 2, …, m} is the hyperarc set.

Ni+ = {j | (i,j) E}

Ni– = {j | (j,i) E}

e: generic hyperarc, with TL(e) N defined as the tail and HD(e) N

defined as the head of the hyperarc. It should be noted that normal arcs are a

sub-set of hyperarcs for which |HD(e)| = 1, where |HD(e)| is defined as the

cardinality of the hyperarc (Nielsen 2004). For reasons of clarity and simplicity,

all the hyperarcs for which |HD(e)| = 1 will henceforth be called ‘arcs’, and

‘hyperarcs’ only those for which |HD(e)| > 1. A generic hyperarc may also be

defined as e = (i, Lip), where i N and Lip Ni+

Hp: hyperpath connecting a single origin destination pair (r,s).

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 8

The following sub-sets of N and E are defined:

CN centroid nodes

PN: pedestrian nodes;

RN: stop nodes;

LN: line nodes.

PE: pedestrian arcs;

LE: line arcs. Each arc e LE is uniquely associated with a line ℓ ℑ;

DE: dwelling arcs;

DuE: dummy arcs, connecting the stop nodes to the pedestrian network

SE: support hyperarcs (Bellei et al. 2000), representing all the lines sharing a

stop. Namely, for the generic stop node i, a support hyperarc a is defined such

that TL(a) = i and HD(a) = Ni+. Each branch b = (i,j) of a support hyperarc is

also called boarding arc (b BE);

WE: waiting hyperarcs (Gentile et al. 2005), representing only the attractive

lines serving the stop. Considering the same stop node as before, the associated

waiting hyperarc c is defined such that TL(c) = i and HD(c) = Lip*() ⊆Ni

+,

where Lip*() represents the set of attractive lines serving the stop node i for

passengers travelling along the hyperpath Hp at time .

AE: alighting arcs.

It should be noted that in the dynamic and congested scenario the branches of

the hyperarc (Figure1) do not only represent the ‘average delay due to the fact that the

transit service is not continuously available over time’, but also the ‘time spent by users

queuing at the stop and waiting that the service become actually available to them’

(Meschini et al 2007). Moreover, for modelling purposes, we assume that passengers

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 9

arriving at the stop join a unique, mixed queue regardless of their particular attractive

line set. In this case, overtaking is possible among passengers having different attractive

sets; however, any competition among passengers sharing the same attractive set is

solved by applying the FIFO rule. (Trozzi et al 2010).

Figure 1: Representation of a stop in the hypergraph.

In order to represent time-dependent travel times, waiting times, etc., the

following dynamic variables are also introduced:

kij(): number of vehicle arrivals that passengers, arriving at stop node i at time

, have to wait before boarding the attractive line associated with the line node j;

tij(): exit time from arc h = (i,j) E for users entering it at time ;

cij(): travel time of arc h = (i,j) E for users entering it at time . Namely, if

the arc is entered at time and the exit time is tij(), then cij() = tij() – ;

ij(): mean frequency at time and stop i of the public transport line

associated with line node j;

ije(): diversion probability. This is the probability of using at time the

boarding arc b = (i,j), which is a branch of hyperarc e = (i, Lip);

ie(): expected waiting time at the stop node i and time , if the considered

hyperarc is e=(i, Lip) ;

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 10

ije(): partial waiting time at the stop node i and time , if the considered

hyperarc is e=(i, Lip) ;

i|je(): expected waiting time at the stop node i and time conditional to the

event of boarding line represented by node j out of the set represented by

hyperarc e=(i, Lip) ;

ti|je(): boarding time for users arriving at stop i at time , conditional to the

event of boarding line represented by node j out of the set represented by

hyperarc e=(i, Lip). Namely: ti|je() = + i|je();

gpis(): expected actual cost of the hyperpath Hp to destination s for users leaving

node i at time ;

Sis(): expected travel cost of the minimal hyperpath to destination s for users

leaving node i at time .

3.2 Problem definition

In the original formulation of shortest hyperpaths (Nguyen and Pallottino 1988, Spiess

and Florian 1989), it is acknowledged that at any intermediate stop, passengers

travelling towards a specific destination would only consider the subset of attractive

lines, which are used in order to minimize the total expected travel time to destination,

and would board whichever attractive line comes first. Consequently, each elemental

itinerary is a particular realisation of the optimal strategy or, from a graphical point of

view, a single path of the shortest hyperpath.

3.2.1 Dynamic shortest hyperpath: definition

A subgraph Hp = (Np, Ep, p()), where Np N, Ep E and p() = (ije()) a real

value vector of dimension (Ep x 1) is a dynamic hyperpath connecting origin r CNp

and destination s CNp, if:

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 11

Hp is acyclic with at least one arc;

node r has no predecessors and node s has no successors;

for every node i Np \ {r, s} there is a hyperpath from r to s traversing i. Each

node has at most one immediate successor hyperarc: if i RNp then its

successor has cardinality equal to one, otherwise the successor hyperarc has

cardinality equal or greater than one;

the elements of the characteristic vector p() satisfy the conditions:

0

,1)(

ije

p

Lj

ije RNiip

(1)

and the value of its components depends on the time they are evaluated;

travel times cij() associated to arcs (i,j) Ep and waitinig times ie()

associated to nodes i RNp depend on the entering time they are evaluated;

waiting times ie() also depend on the considered hyperarc e = (i, Lip()) Ep.

In the static context, it is shown that the total travel time of the generic

hyperpath Hp can be computed by explicitly taking into account all the elemental paths ℓ

forming it (Nguyen and Pallottino 1988, 1989). Therefore, if Qp is the set of such paths,

ℓ is the probability of choosing the elemental path ℓ, and ℓ is its travel time, the travel

time of hyperpath Hp is:

pQ

pg

(2)

On the other hand, ℓ can be expressed as the sum of travel and waiting times on

the path’s arcs and nodes:

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 12

pp NRi

ii

Eji

ijijc ',

(3)

where ijℓ = 1 if arc h = (i,j) belongs to path ℓ, otherwise ijℓ = 0, and 'iℓ = 1 if

path ℓ traverses node i, otherwise 'iℓ = 0. Thus the following expression of the

hyperpath’s total travel time can be obtained:

p ppQ RNi

ii

Eji

ijijp cg

',

(4)

In a network with overcrowding, as considered here, travel times depend on the

time the arc is entered. Consequently, it can happen that the same node is traversed by

different paths at different times and the travel cost associated with it has different

values. Hence, the above definition of the hyperpath’s total travel time does not apply to

the dynamic scenario. Nevertheless, by extending the local recursive formula (the so-

called generalised Bellman equation) given in Nguyen and Pallottino (1988, 1989) to

the dynamic context, a sequential definition of the dynamic hyperpath’s travel time

structure is obtained. What is more, the equation enables obtaining the result whilst

avoiding path enumeration, thus decreasing the computational burden.

The total travel time of the dynamic hyperpath Hp connecting r to s at time is

sequentially defined in reverse topological order as:

peiHD

seHD

pie

pij

js

pij

is

p

RNitg

RNitgc

si

g

if ,)()(

if ,)()(

if ,0

)(

)(

(5)

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 13

p

Lj

jei

js

pijeie

pij

js

pij

is

p

RNitg

RNitgc

si

g

ip

if ,

if ,)()(

if ,0

|

(5)b

where:

ie() is the total expected waiting (and queuing) time at stop i associated with

hyperarc e = (i, Lip) ;

gpHD(e)s

(tiHD(e)( )) is the remaining cost of the hyperpath, upon boarding one of

the attractive line(s) from stop node i;

Lip is the particular set of competing lines, that passengers might consider

boarding at stop i when travelling along hyperpath Hp.

3.3 Stop model

3.3.1 Probability distribution functions of the waiting times

It should be noted that the above formulation of the hyperpath travel time structure and

computation is independent of specific values given to the diversion probabilities ije(),

and expected waiting costs ie(). These variables are specified by the stop model and

depend on the particular assumptions adopted for modelling the passenger and the

public transport vehicle arrivals at the stop node, and on the passenger boarding

mechanism.

In our model, the basic hypotheses about carrier and passenger arrivals (Nguyen

and Pallottino 1988, Spiess and Florian 1989) are not changed, but passengers waiting

at a stop may be prevented from boarding an approaching carrier because of

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 14

overcrowding. If the stop layout is such that passengers have to join a FIFO queue, then

the user coming at stop i at time has to wait for the kij()-th

carrier of line j.

As proved by Larson and Odoni (1981, p. 54), the waiting time before the kij()th

carrier arrival occurs is Erlang-distributed with parameters kij() and ij(),

otherwise 0,

0 if ,!1)(

)(exp)(

),(

1)()(

wk

ww

wfij

k

ij

jk

ij

ij

ijbi

(6)

where w is the stochastic variable representing the waiting time.

As such, considering hyperarc e = (i, Lip()), the diversion probability can be

expressed by the following formula:

dwwFwfjLz

izijije

ip

0 \

),(),()( (7)

where F iz(w,) is the survival function of the Erlang-distributed waiting time for the

branch b=(i, z) of hyperarc e = (i, Lip).

On the other hand, the total expected waiting time for the considered stop i and

hyperarc e is equal to:

dwwFipLj

ijie

0

),()( (8)

Recalling the definition of conditional expected value (Loève 1978, Melotto

2004), it is also possible to write:

ipLj

jeiijeie )()()( | (9)

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 15

dwwFwfwjLz

izij

ije

jei

ip

0 \

| ),(),()(

1)(

(10)

To further stress the implication of the proposed stop model, consider the small example

network of Section 2 and the four scenarios summarised in Table IV.

Scenario 2 is the same considered in Section 2. As displayed in Table V, the

probability to board Line 1 is not only lower than the value calculated with the

uncongested model (Table II), but also lower than the value calculated with the

effective frequency model (Table III). On the other hand, the opposite holds for the total

expected waiting time.

Table IV: Average frequencies, k values and travel time upon boarding for the two lines

in the considered scenarios.

Line 1

Average

frequency

[min-1

]

Line 1

k

Line 1

Travel time

upon boarding

[min]

Line 2

Average

frequency

[min-1

]

Line 2

k

Line 2

Travel time

upon boarding

[min]

S 1 1/6 1 10 1/6 1 10

S 2 1/3 2 10 1/6 1 10

S 3 1/2 3 10 1/6 1 10

S 4 1 6 10 1/6 1 10

Table V Boarding probabilities, conditional expected waiting times and total expected

waiting time at the stop for the considered scenarios.

ije() θi|je() [min] θie() [min]

S 1

Line 1 0.50 3.00 3.00

Line 2 0.50 3.00 3.00

S 2

Line 1 0.45 3.99 3.33

Line 2 0.55 2.81 3.33

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 16

S 3

Line 1 0.42 4.50 3.47

Line 2 0.58 2.72 3.47

S 4

Line 1 0.40 5.13 3.62

Line 2 0.60 2.62 3.62

This phenomenon may be explained with the properties of the Erlang and

exponential distributions. In fact, if k = 2 and = 1/3 min-1

for Line 1, this service

would be boarded only in case both vehicles of Line 1 pass with an headway shorter

than the average value of three minutes. However, this event is less probable than one

vehicle of Line 2 arriving before its average inter-arrival time (six minutes). Moreover,

if the mean of the Erlang distribution is constant (in our case, k/ = 6) and k → ∞, the

waiting time before boarding tends to be deterministic, thus θi|je() for Line 1 increases

and tends to kline1/line1.

On the other hand, the boarding probability for Line 2 tends to equation (11),

dwwf

linek

lineline line1

1

0

22 ),()(

(11)

where, kline1/ line1 = 1/ line2 is also the expected value of fline2(w, ). Because the

expected value of an exponential distribution is always lower than its median, the

boarding probability of Line 2 progressively increases with k line1 (for constant k line2 =

1), while π line1 decreases.

3.3.2 Attractive Set

In general, the above expressions of the diversion probabilities and expected waiting

times can be applied to any subset of Ni+, however, only a specific subset L

*ip() Ni

+

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 17

is associated with the minimum travel time to destination at time . This set is defined

attractive and includes all the lines nodes that make up the head of the waiting hyperarc

associated to the stop node considered.

Recalling the definition of an attractive set given in (Nguyen and Pallottino

1988), we can write:

ipipiip

ipip

Lj

jei

js

pije

Lj

ijeNL

Lj

jei

js

pije

Lj

ije

iip

tS

tg

NL

)()()(min

)()()(

:)(

|

)(

|

)(

*

**

(12)

Consequently, in order to determine L*

ip(), it is in general necessary to compute

gis

p() for all the possible subsets of Ni+. However, at least for the uncongested static

case, it is counter-intuitive to exclude a line from L*

ip() if it has a shorter remaining

travel time than any other attractive one. Therefore, it is possible to solve the

combinatorial problem described above through a greedy approach (Spiess and Florian

1989, Nguyen and Pallottino 1988, Chriqui and Robillard 1975). Namely, the lines are

processed in ascending order of their travel time upon boarding and the progressive

calculation of the values of ije, ie and gis

p is stopped as soon as the addition of the next

line increases the value of gis

p. At this point, the cost is minimal (Sis) and the set of lines

corresponds to the attractive set.

The correctness of the greedy method, in the static case, depends on the shape of

the waiting time pdf (exponential). By contrast, in the dynamic scenario the only exact

method of finding L*

ip() requires the enumeration of all the possible combination of

lines serving the considered stop. Greedy-type heuristics could be applied also in the

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 18

time-dependent case to overcome the computational complexity arising from the exact

solution of the problem, however this is out of the scope of the present work.

4. Numerical Example

The Decreasing Order of Time (DOT) method, presented by Chabini (1998), which has

been analytically proved to be the most efficient solution method for the all-to-one

search for every possible arrival time, was extended to the time-dependent shortest

hyperpath problem in order to devise a solution algorithm for the example considered in

this section.

Although the proposed model has a continuous time representation, a discrete-

time representation for its numerical solution has been adopted. The main idea is to

divide the analysis period AP = [0, Θ] into T time intervals, such that AP =

{T-1}, with = 0 and T-1

= Θ, and to replicate the network along the

time dimension, forming a time-expanded hypergraph HGT containing vertexes in the

form (i, ), and edges in the form ((i, ), (j, tij())).

If time intervals are short enough to ensure that the exit time of a generic edge

tij() is not earlier than the next interval , for T-2, it is ensured that the network

is cycle-free and the vertex chronological ordering is equivalent to the topological one.

Thus, HGT is scanned starting from the last temporal layer to the value assumed for =

and, within the generic layer, no topological order is respected. When a generic

vertex (i, ) is visited, its forward star is scanned in order to set the minimal travel cost

to destination and the successive edge by means of equation (12) (refer to the appendix

for a detailed formulation of the algorithm).

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 19

Such algorithm has been applied to the network used by Spiess and Florian

(1989) in their seminal work (Figure 2) in order to find optimal travel strategies and

calculate travel times from each node to destination (node 16).

The results obtained in the static and uncongested scenario, with reference to

destination node 16, are summarised in Figure 3: for each stop the waiting hyperarc is

depicted, Si indicates the total expected travel time from node i to destination and i-j

is

the probability to traverse the boarding arc (i, j).

Figure 2: Example network, the graphic representation is consistent with Figure 1.

Figure 3: Optimal strategy found for the static case, expected total travel time from each

intermediate node to the destination and diversion probabilities at the stop nodes.

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 20

In order to compare static conditions with results obtained when hypothesising

time-dependence of passenger congestion and travel variables within the day, the

morning peak [08:00 –9:30] is divided into one-minute intervals. Changes in travel

conditions are assumed to occur only at 08:00, 08:30, and 09:00, as shown in Table VI,

while outside the analysis period travel variables are assumed to remain constant and

have the same values as in the uncongested scenario.

Table VI: Time dependent travel variables for each line arc of the example network: in-

vehicle travel time, average frequency and number of passages to be waited before

boarding (k).

Time of

the day

Travel variable Arc 3

(4,13)

Arc 4

(5,6)

Arc 9

(7,9)

Arc 10

(8,10)

Arc 16

(11,14)

Arc 17

(12,15)

8:00-08:30 Travel time [min] 30 7 6 4 4 15

8:00-08:30 Freq. [min-1

] 1/3 1/6 1/6 1/15 1/15 1/3

8:00-08:30 k 2 1 1 1 2 1

8:30-09:00 Travel time [min] 35 7 6 4 4 15

8:30-09:00 Freq.[min-1

] 1/3 1/6 1/6 1/15 1/10 1/3

8:30-09:00 k 3 1 1 1 2 2

9:00-09:30 Travel time [min] 35 7 6 4 4 10

9:00-09:30 Freq. [min-1

] 1/3 1/6 1/6 1/15 1/5 1/3

9:00-09:30 k 2 1 1 1 2 1

Results, summarised in Figures 4-5-6, show that the change in travel variables

does not only affect the expected travel time to the destination and the boarding

probability, but also the alternatives included in the optimal strategy. Notably, between

08:00 and 08:30, because of congestion at stop 3, passengers waiting at stop 2 do not

include Line 001 in their attractive set. Instead, they prefer boarding Line 003 so as to

avoid a transfer at stop 3 and the associated long waiting time. Also, between 08:30 and

09:00, because of the increased travel time upon boarding and heavy (k = 3) congestion,

Line 002 is excluded from the attractive set at stop 1 and is not even re-included after

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 21

09:00, when congestion slightly decreases (k = 2). Finally, between 09:00 and 09:30,

the total expected waiting time at stop 3 remarkably decreases, because Line 004

becomes more frequent and congestion on Line 003 has dissipated. Consequently,

passengers waiting at stop 2 now choose to make a transfer at stop 3 and both lines (001

and 003) are included in the attractive set.

Figure 4: Optimal strategy found between 08:00 and 08:30, expected total travel time

from each intermediate node to the destination and diversion probabilities at the stop

nodes.

Figure 5: Optimal strategy found between 08:30 and 09:00, expected total travel time

from each intermediate node to the destination and diversion probabilities at the stop

nodes.

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 22

Figure 6: Optimal strategy found between 09:00 and 09:30, expected total travel time

from each intermediate node to the destination and diversion probabilities at the stop

nodes.

5. Conclusions

Given the importance of travel time variability in the traveller route and mode

decisions, this paper presents a dynamic route choice model for public transport

networks, in which travel variables, such as on-board travel times and frequencies, vary

with time, and in which queues may form at public transport stops due to overcrowding,

hindering passengers to board the first available line of their choice.

The model reproduces route choice of commuting passengers that know by previous

experience the number of passages they must let go, at each stop, before being able to

board every line of their attractive set. In case of constant variables, the results from the

static algorithm by Spiess and Florian (1989) are reproduced. However changes in the

shortest hyperpath are clearly shown when travel variables become dynamic and/or

queuing at stops is considered.

In the example given, the combinatorial problem of selecting, at each stop, the

set of attractive lines, is solved by means of an exact procedure. However, when

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 23

applying the model in large-scale scenarios, using real network and traffic variables

data, heuristics are needed to speed up the computation.

Although the expected travel time is an important factor affecting the travellers’

route choice, as proved by the numerical example, many studies have found that the

reliability of travel time due to non-recurrent congestion can be even more important.

Indeed a “wrong” choice of route or mode can result in a significant delay, which may

have severe consequences for the traveller (e.g. late arrival at the workplace). Equally,

some groups of travellers may be particularly adverse to discomfort on board due to

overcrowding (e.g. elderly travellers, parents travelling with your children …) and

decide to change their route and/or departure time according to network conditions. As

such, future developments will concentrate in the inclusion of these factors in a multi-

class route choice model coupled with departure time choice model in public transport

networks.

Also, within the same context, the method will be tested in large-scale scenarios,

using real network and traffic variables data, and its potential of application to real-time

journey planning will be investigated.

6. Appendix: the solution algorithm

The variable list of the algorithm is:

: temporal layer index

Int: temporal layer length in minutes

s: destination node

i: generic node

FS(i): set of (hyper)arcs belonging to the forward star of node i

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 24

a: generic hyperarc, where a FS(i)

b: branch (i,j) of the generic hyperarc a FS(i), where i RN

c(i,): waiting hyperarc from stop node i at time interval .

HD(c(i,)) = Lip*()

suc(i,):successor arc of the generic node i at time interval , where i RN

h: generic arc, where h FS(i) and i RN

gis(): current travel cost from generic node i to destination s at time interval

Sis(): minimum travel cost from generic node i to destination s at time interval

Sis

stat: minimum travel cost from generic node i to destination s at time interval

gis(j,a,): travel cost from i to s if passengers travel along branch b=(i,j) of

hyperarc a at time interval

The solution algorithm for the time-dependent all-to-one shortest hyperpath

problem for every possible arrival time is detailed here.

Step 0 (Static pre-processing – Initialisation): ∀ i N \ {s}

Calculate Sis() = S

isstat

∀ [0,T -2]

Set Sss

() = 0, suc(s,) =

∀ i N \ {s}

Set Sis() =

Step 1 (Hyperarcs’ dynamic attributes): ∀ [0,T -2]

∀ i RN

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 25

∀ a FS(i)

∀ b = (i,j) ⊆ a

Calculate ija() with eq. (7)

Calculate ija() with eq. (11)

ia() =ia(

) + ija(

)

Step 2 (Calculate hyperpath travel time): ∀ [0,T-2]

∀ i N \ {s}

If i RN, ∀ a FS(i)

∀ b = (i,j) ⊆ a

If [[i|ja() / Int]] > 1

ti|ja() = [[i|ja(

) / Int]] +

Else

ti|ja() = + 1.

If ti|ja() < and

gj(ti|ja(

)) <

gis(j,a,) = g

j(tija(

))

Else

gis(j,a,) = S

jsstat.

gis() = g

i() +

gis(j,a,).ija(

)

If Sis() > gp

is() then

Sis() = gp

is() And c(i,) = a

Else if i RN, ∀ h FS(i)

If [[ cij() / Int]] > 1

V. Trozzi, I. Kaparias, A dynamic route choice model

M. G. H. Bell and G. Gentile for public transport networks with boarding queues

Transportation Planning and Technology, DOI: 10.1080/03081060.2012.745720 26

tij() = [[ cij(

) / Int]] +

Else

tij() = + 1

End if

gis() = cij(

) + gjs(tij(

))

If Sis() > g

is()

Sis() = g

is() and

suc(i,) = h

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